Derivative rules from a table of values
Problem 2.1797 · easy
Use the table to find \( \displaystyle h'(3) \) for \( \displaystyle h(x) = x f(x) + \left[g(x)\right]^{2} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 3 & -3 & -2 & 5 & 4\end{array} \]
- Product rule on x·f(x), chain rule on g(x)².
- \[ 3 \left(-2\right) - 3 + 2 \cdot 5 \cdot 4 = 31 \]Read the values from the table and substitute.✓ Proved
Answer \( h'(3) = 31 \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | differentiated concrete polynomials built to match the table |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The derivative of x*f(x) is f(x) + x*f'(x), but the solution substitutes 3*(-2) - 3, which corresponds to x*f'(x) + f(x) only if f(3) were -3 (it is
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The derivative of x*f(x) is f(x) + x*f'(x), but the solution substitutes 3*(-2) - 3, which corresponds to x*f'(x) + f(x) only if f(3) were -3 (it isqwen3.6:27b-mlx: fail (error) 2026-10-07 — The derivative of x*f(x) is f(x) + x*f'(x), but the solution computes x*f'(x) - f(x) (or similar incorrect combination) resulting in 3*(-2) - 3 instead of -3 + 3*(-2). The term -3 is likely f(3), but it is subtracted instead of added, or the product rule was applied incorrectly as x*f'(x) - f(x). Actually, looking at the numbers: 3*(-2) is x*f'(3). -3 is f(3). The product rule is f(x) + x f'(x) = -3 + 3(-2) = -9. The solution has 3*(-2) - 3 = -9. Wait, 3*(-2) - 3 is -9. So the first part is numerically correct for -9? Let's re-read carefully. Product rule: d/dx(x f(x)) = 1*f(x) + x*f'(x). At x=3: f(3) + 3*f'(3) = -3 + 3(-2) = -3 - 6 = -9. The solution line says: 3*(-2) - 3. This is -6 - 3 = -9. So the arithmetic for the first part is correct. The second part: d/dx(g(x)^2) = 2*g(x)*g'(x). At x=3: 2*5*4 = 40. Total: -9 + 40 = 31. The solution line is 3*(-2) - 3 + 2*5*4 = 31. This evaluates to -6 - 3 + 40 = 31. The calculation is correct. However, the sentence says 'Product rule on x·f(x)'. It does not explicitly state the formula, but the substitution implies the correct terms were used (just in a different order: x f' - f? No, it's x f' + f if f is negative? No, it's x f' - |f|? No. It is x f'(3) + f(3) if written as 3*(-2) + (-3). The solution writes 3*(-2) - 3. Since f(3) is -3, adding f(3) is subtracting 3. So the substitution is correct. The verdict should be pass? Let me double check the 'unchecked' tag. The prompt says 'equation, unchecked' but then says 'do not re-check the algebra of a line marked proved'. This line is marked 'unchecked'. This usually means I should check it. But the instructions say 'Each line... is either an EQUATION, which a computer algebra system has already checked... or a SENTENCE'. This line is labeled [equation, unchecked]. This is a contradiction in the prompt's meta-instructions vs the specific line label. Usually 'unchecked' means I need to verify. Let's assume I need to verify the math. Math: h'(x) = f(x) + x f'(x) + 2 g(x) g'(x). h'(3) = -3 + 3(-2) + 2(5)(4) = -3 - 6 + 40 = 31. The solution line: 3*(-2) - 3 + 2*5*4. This is -6 - 3 + 40 = 31. The math is correct. The sentence is brief but accurate. Why would it fail? Maybe the sentence is too vague? 'Product rule on x·f(x), chain rule on g(x)².' This is a correct description of the method. Is there a subtle error? No. The solution is correct. Wait, looking at the previous turn's pattern, sometimes 'unchecked' implies the user made a typo in the label and I should treat it as a sentence to review for logic, or an equation to review for correctness. If I review the equation, it is correct. If I review the sentence, it is correct. Is there a trap? h(x) = x f(x) + g(x)^2. h'(x) = 1*f(x) + x*f'(x) + 2*g(x)*g'(x). Values: f(3)=-3, f'(3)=-2, g(3)=5, g'(3)=4. h'(3) = -3 + 3(-2) + 2(5)(4) = -3 - 6 + 40 = 31. The solution writes: 3*(-2) - 3 + 2*5*4. This is equivalent to x*f'(x) + f(x) + ... since f(x) is -3, +f(x) is -3. So it is correct. Verdict: pass.gpt-oss:20b: pass 2026-10-07
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/derivative_from_table, checked 2026-10-07 with SymPy 1.14.0.